RICE UNIVERSITY Diagnosing the Frequency of Energy Deposition in the Magnetically-closed Solar Corona b y Will Barnes A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Doctor of Philosophy APPROVED, THESIS COMMITTEE: o^y^r^y^i^o Dr. Stephen Bradshaw, Chair Associate Professor of Physics and Astronomy ^ v -Dr. David Alexander, OBE Professor of Physics and Astronomy )r. Maarten V. de Hoop Simons Chair and Professor of Computational and Applied Mathematics and Earth Science HOUSTON, TEXAS MAY 2019 ABSTRACT Diagnosing the Frequency of Energy Deposition in the Magnetically-Closed Solar Corona by Will Barnes The solar corona, the outermost layer of the Sun’s atmosphere, is heated to tempera- tures in excess of one million Kelvin, nearly three orders of magnitude greater than the surface of the Sun. While it is generally agreed that the continually stressed coronal magnetic field plays a role in producing these million-degree temperatures, the exact mechanism responsible for transporting this stored energy to the coronal plasma is yet unknown. Nanoflares, small-scale bursts of energy, have long been proposed as a candidate for heating the non-flaring corona, especially in areas of high magnetic activity. However, a direct detection of heating by nanoflares has proved difficult and as such, properties of this proposed heating mechanism remain largely unconstrained. In this thesis, I use a hydrodynamic model of the coronal plasma combined with a sophisticated forward modeling approach and machine learning classification techniques to predict signatures of nanoflare heating and compare these predictions to real observational data. In particular, the focus of this work is constraining the frequency with which nanoflares occur on a given magnetic field line in non-flaring active regions. First, I give an introduction to the structure of the solar atmosphere and coronal heating, discuss the hydrodynamics of coronal loops, and provide an overview of the important emission mechanisms in a high-temperature, optically-thin plasma. Then, I describe the forward modeling pipeline for predicting time-dependent, multi-wavelength emission over an entire active region. Next, I use a hydrodynamic model of a single coronal loop to predict signatures of “very hot” plasma produced by nanoflares and find that several effects are likely to affect the observability of this direct signature of nanoflare heating. Then, I use the forward modeling code described above to simulate time-dependent, multi-wavelength AIA emission from active region NOAA 1158 for a range of nanoflare frequencies and find that signatures of the heating frequency persist in multiple observables. Finally, I use these predicted diagnostics to train a random forest classifier and apply this model to real AIA observations of NOAA 1158. Altogether, this thesis represents a critical step in systematically constraining the frequency of energy deposition in active regions. Acknowledgements This work would not have been possible, or at the very least far less enjoyable, without the help and support of my supervisors, colleagues, friends and family. First, I would like to thank my thesis committee members, Dr. David Alexander and Dr. Maarten de Hoop, for agreeing to serve on my committee and reading a first draft of this work. I would especially like to thank Dr. Alexander for his advice and guidance, both career- and research-related, during my time as a graduate student and for helping me navigate the field of solar physics. As a graduate student, it has been my great pleasure to work with and be advised by Dr. Stephen Bradshaw. I have benefited immensely from his vast knowledge of field-aligned hydrodynamics and atomic physics as well as his careful and measured approach to research. Most importantly, he has taught me how to be an independent researcher and I am extremely grateful for his mentorship and friendship during my time at Rice. I also owe a special debt of gratitude to my undergraduate research advisor, Dr. Lorin Matthews (Baylor University), for teaching me about the microphysics of astrophysical dusty plasmas and for inspiring me to go to graduate school. I am grateful for her patience and kindness as a mentor early in my physics education. During my brief time in the solar physics community, I have been fortunate to collaborate with several talented and accomplished researchers. I am extremely grateful to Professor Peter Cargill (Imperial College London, University of St An- drews) for sharing his unparalleled knowledge of coronal loop physics and for v his patience in guiding me through the writing and publication of two papers early in my graduate career, the first of which comprises Chapter 5 of this thesis. Additionally, I would like to thank Dr. Nicholeen Viall (NASA Goddard Space Flight Center) for lending her observational expertise and detailed knowledge of the temperature sensitivity of the AIA passbands and for providing helpful comments and revisions on Chapter 6 and Chapter 7 of this thesis. I would also like to thank Dr. Jim Klimchuk, Dr. Harry Warren, Dr. Jeffrey Reep, Dr. Jack Ireland, and Dr. Ken Dere. I am extremely indebted to the members of the SunPy community for volunteer- ing their time and effort to build a sustainable software ecosystem for solar physics. In particular, I would like to thank Dr. Stuart Mumford for his tireless and often thankless efforts to continually improve and develop SunPy and for always having the answer to any question related to Python or solar coordinate systems. I am very grateful to the many people at Rice and in Houston who made my time as a graduate student all the more enjoyable. Many thanks go to Dan, Kong, Joe, Nathan, Loah, Brandon, Laura, Alison, Alex, and Shah for hearing my complaints at lunch, sharing more than a few beers at Valhalla, and making graduate school bearable and, on occasion, fun. I would especially like to thank Joe and Mitch for their friendship and support over the last decade, both in Waco and in Houston. I would like to thank my parents, Mark and Terri, for their financial, emotional, and physical support throughout my entire life, across multiple states and even a few continents. I would also like to thank my siblings, Jessie and Wesley, for always being willing to remind me that I am not that smart. I owe special thanks to my in-laws, Jim and Susan, as well as to my siblings-in-law, Tara and Michael, for the many rounds of disc golf and even more free meals; and Tamara and Mike for making me feel welcome when I first moved to Houston and for continuing to support me as I prepare to leave. vi Lastly and most importantly, I am forever grateful to my wife Morgan, to whom this thesis is dedicated. Her ever-present optimism, constant encouragement, and unmatched love of dogs have made life all the more enjoyable, even in the face of looming deadlines. Without her unconditional love and support, I would not have made it to graduate school let alone finished this thesis. The Sun is a miasma Of incandescent plasma The Sun’s not simply made out of gas No, no, no —“Why Does the Sun Really Shine” They Might Be Giants Table of contents List of figures xiii List of tables xxx Nomenclature xxxii 1 Introduction 1 1.1 The Structure of the Solar Atmosphere . 2 1.1.1 Interior . 3 1.1.2 Photosphere . 5 1.1.3 Chromosphere . 7 1.1.4 Transition Region . 7 1.1.5 Corona . 8 1.1.6 The Solar Wind . 9 1.2 The Solar Magnetic Field . 10 1.2.1 Origin of the Magnetic Field and Flux Emergence . 11 1.2.2 Observations . 14 1.2.3 Field Extrapolation . 15 1.2.4 Reconnection . 17 1.3 Heating in the Solar Corona . 20 1.3.1 Waves versus Reconnection . 21 Table of contents ix 1.3.2 Nanoflare Heating . 23 1.4 Thesis Outline . 26 1.5 Use of Data and Software . 28 2 The Physics of Coronal Loops 29 2.1 Hydrostatics . 32 2.1.1 Equations of Hydrostatic Equilibrium . 32 2.1.2 The Isothermal Limit . 36 2.1.3 Scaling Laws . 39 2.1.4 Numerical Solutions . 44 2.2 Hydrodynamics . 46 2.2.1 Equations of Field-aligned Hydrodynamics . 47 2.2.2 The Heating, Cooling, and Draining Cycle of Coronal Loops . 50 2.2.3 The HYDRAD Model . 54 2.2.4 The EBTEL Model . 55 3 Emission Mechanisms and Diagnostics of Coronal Heating 67 3.1 The CHIANTI Atomic Database . 68 3.2 Spectral Line Formation . 69 3.2.1 Collisional Excitation of Atomic Levels . 70 3.2.2 Level Populations . 75 3.2.3 Processes which Affect the Ion Charge State . 76 3.2.4 The Charge State in Equilibrium . 82 3.2.5 Non-Equilibrium Ionization . 84 3.3 Continuum Emission . 87 3.3.1 Free-free Emission . 87 3.3.2 Free-bound Emission . 88 3.4 Temperature Sensitivity of the AIA Passbands . 90 Table of contents x 3.5 The Differential Emission Measure Distribution . 94 3.5.1 The Emission Measure Slope . 96 3.5.2 Determining the DEM from Observations . 98 3.6 Time-Lag Analysis . 104 3.6.1 Cross-Correlation . 105 3.6.2 Time Lag between AIA Channel Pairs . 107 4 synthesizAR: A Framework for Modeling Optically-thin Emission 111 4.1 Building the Magnetic Skeleton . 112 4.1.1 Potential Field Extrapolation . 113 4.1.2 Tracing Magnetic Field Lines . 116 4.1.3 Aside: Coordinate Systems in Solar Physics . 117 4.2 Field-Aligned Modeling . 122 4.3 Atomic Physics . 124 4.4 Instrument Effects . 126 4.4.1 Constructing the Virtual Observer .